Craig Reid, recalled to Boro's XI, lashed home his 21st goal of the season after 24 minutes to set the hosts on their way to a third straight home win before Joel Byrom's contentious penalty and Chris Beardsley's late strike rubber-stamped sixth place.
Stevenage, aiming for a third successive promotion, will now face Sheffield United in the play-offs.
After a tense opening, the hosts broke the deadlock when Luke Freeman whipped in a dangerous cross which Bury defender Efe Sodje could only flick into the direction of Reid, who chested the ball down and lashed home.
Shakers goalkeeper Jonathan Bond produced a fine stop to deny Freeman and soon after the break he was at it again, thwarting substitute Robin Shroot and Lawrie Wilson in quick succession.
Bryom then made it two-nil from the spot after 77 minutes when Sodje was adjudged to have brought down Shroot, before Beardsley wrapped things up from the edge of the area late on.